Generation of Ultrashort XUV Pulses with High-Harmonic-Generation

High-harmonic generation (HHG) is an optical non linear process, that converts light with a fundamental frequencyω₀to light of a high multiplenω₀(n >10) of the fundamental.

Under non-resonant condition every order of non-linear polarizationP⁽ⁿ⁾∝Eⁿbecomes smaller by a factor∼ ^E/E_at, whereE_at is the characteristic atomic field of the material², i.e. higher harmonics become weaker by their order and the lowest orders dominate the whole optical response and can be realistically used for frequency conversion. In the late 1980s [MGJ⁺87] it was discovered that noble gases upon irradiation by high intensity (I > 10¹⁴ W cm⁻²) ultra short pulses emit very high harmonics up to the 21^st order [FLL⁺88]. These first experiments already outlined the typical spectral shape of HHG. Only odd harmonics are visible as well defined peaks and the intensity decreases

2For simplicity the characteristic atomic field is often taken to be the Coulomb field of the hydrogen at one Bohr radius distance, which yieldsE_at≈0.51 kV nm⁻¹.

for the first few orders but forn≳5 reaches a plateau of almost constant intensity up to a sharp cutoff.

The general physics of the process was soon understood and theoretical models that could reproduce the general shape of the experimental spectra were developed. Corkum pub-lished the so called three step model in 1993 [Cor93] that despite its extreme simplicity captures the most important features . The three step model divides HHG into three separate processes, namely ionization, acceleration in the vacuum and recombination.

In the first step the bound electron tunnels through the Coulomb barrier, which is dis-torted by the linear external potential. In his original paper Corkum uses a semi classical (WKB) tunneling model but other methods such as direct solutions to the time-dependent Schroedinger equation (TDSE) are possible as well. After the ionization a classical treat-ment of the electron is used and the electron is assumed to be at the atomic position with zero velocity directly after ionization. The coulomb interaction is neglected in this step.

This leads to the extremely simple equation of motion r¨e = − e

m_eE₀cos(ω₀(t−t₀))

which can be solved analytically. The maximum kinetic energy the electron can extract from the external field is

E_kin,max =3.17U_p U_p= e²E²₀

4m_eω²₀

with the ponderomotive potential U_p. The extracted kinetic energy depends on the phase of the field with respect to the point in time of ionization and is optimal around

16π. Smaller phases lead to a shorter excursion time and distance and higher phases to longer excursion time and distance, both however with smaller kinetic energy. These two types of trajectories are often simply labeled short and long trajectories. The last step, recombination, occurs when the electron returns to the nucleus. Due to the high kinetic energy of the electron this occurs on an attosecond timescale and the process emits an attosecond pulse, which is extremely broad and continuous in frequency space. The process with the maximal emitted photon energy is when the electron recombines with the ion. Then it emits the ionization potentialI_pin addition to the kinetic energy, so the cutoff is

E_cutoff=3.17U_p+I_p

More advanced models solve the time-dependent Schroedinger equation (TDSE), typically in the dipole approximation with

ih∂ _t|ψ⟩= (p²

2m+V(r) −eEr )

|ψ⟩

within various levels of approximation such as the strong field approximation (SFA). The emitted harmonics calculated with the transient dipole

d(t) = −⟨ψ|er|ψ⟩

and the HHG spectrumS(ω)as

S(ω)∝|d(ω)|²

So to recap the general physics of HHG. An electron is extracted from its atom by strong field ionization and accelerated by the field into the vacuum. After the field switches sign it is accelerated back into its atom, where it arrives with a kinetic energy gain, which depends on the phase of the field at extraction. These excess kinetic energy is emitted as light and is responsible for HHG. The emitted spectrum of a single cycle is continuous.

In a multi period driving pulse the interference between HHG from different half periods gives rise to the peaks at uneven harmonics. IfE_s(t)is the field from a single half cycle andA_nis its amplitude, the full field is given by

E_tot(t) = to interference between the half cycles. The(−1)ⁿfactor leads to constructive interference only between odd harmonics. For a simple rectangular intensity profileA_n={1 for 0<

n⩽N, 0 else one can solve the sum analytically and find

|f(ω)|²= sin²((N+1)x)

which peaks at odd harmonics and the peaks show a spectral widthΓ ∝ ¹/N, i.e. the shorter the driving pulse is in time the broader the HHG peaks become. A more realistic HHG spectrum with an Gaussian intensity profile is shown in Fig. 3.3. The qualitative behavior is however quite similar. A single atom principally emits into the full solid angle with a dipole angular dependence. Experimentally we observe an exclusive emission into forward direction. The reason is that for a macroscopic gas cell a coherent emission is necessary in order to effectively radiate light. This condition is called phase matching and quantitatively the coherence length

L_coh= π

|∆k|

must be larger than the gas cell.∆kis the wave vector mismatch between laser light and the respective HHG mode of orderγas

∆k=k_hhg−γkl

15 23 31 39 47 55 63 71 79 Harmonic order

HHGspectrum

∆t= 2 fs

∆t= 5 fs

∆t= 10 fs

Figure 3.3: Sketch of the qualitative behavior of a HHG spectrum with a Gaussian envelope (forAn), calculated with Eq. (3.2). For a higher time duration of the envelope the peaks become successively sharper. Note that the time duration of the envelope is functionally related but not equal to the time duration of the driving pulse due to the non linear nature of the HHG process.

The direction of the HHG’s k vectork_hhgis in principle free and emission occurs in those directions wereL_cohis smaller than the gas cell length. The phase matching condition can only be strictly zero in forward direction. If we assume that it is fulfilled in forward direction, i.e.k_hhg=γkl, we find that for a small angular mismatchϑ

|∆k|≈ϑγ|kl|≈ϑγ2π λ_l The phase matching condition for the angle now reads

L_coh= π

|∆k| = λ_l

2γϑ > L_cell ϑ < λ_l

2γL_cell

We see that the higher the harmonic order and the longer the gas cell, the smaller the cone of emission. For a typical configuration with a Ti:Sapphire laser around 800 nm and a 5 mm gas cell the maximum angle of emission for the 30^thharmonic is 1.5×10⁻⁴°, which means that the HHG beam has basically the same divergence as the original beam.

In forward direction the phase matching condition can be expanded as δk= ω_hhg

c ∆n_at+ ω_hhg

c ∆n_el+k_geo+k_dip (3.3) with the contributions

atomic refractive index ∆n_at=n(ω_hhg) −n(ω_l)

plasma refractive index ∆n_pl=n_pl(ω_hhg) −n_pl(ω_l)

∆n_atis the difference refractive index of the neutral gas between hhg and fundamental frequency. The plasma refractive index contribution∆n_plis caused by ionized part of the neutral gas. The high intensity ionizes a fraction of the gas in the cell, which has a qualitatively and quantitatively different refractive index. The last two terms are the Gouy phase, which is a consequence of the focusing geometry, which is used in most experiments, and the nonlinear dipole phase, which occurs because the emission phase from the three step HHG process depends on the intensity.

Experimentally the gas pressure, intensity of the beam and focus position in relation to the cell can be changed in order to achieve phase matching, therefore the dependence of each term on these parameters is important. For phase matching, i.e.δk=0 a set of conditions has to be found so that all terms cancel each other out. One can typically distinguish two regimes. For relatively low energy harmonicsE < 100 eV, low pressurep < 100 mbar, relatively small pulse energy and a short focus geometry the last terms dominate and can be tuned to cancel each other out. The geometric (Gouy) phase is always positive so the nonlinear phase must be negative, which implies decreasing intensity, i.e. the cell must be behind the focus.

The other extreme case is a relatively high pressure, high pulse energy and long focus geometry, where the first two terms dominate. The first term is always negative because the refractive index at the hhg frequency is very close to one and can be approximated by

∆n_at≈1−n(ω_l) =χ_atρ_at

whereχ_atis the susceptibility per density andρ_at is the density of the gas. The plasma contribution is always positive and can be approximated by

∆n_pl= ω²_pl

2ω²_l = ρ_ple² 2ε₀m_eω²_l

Phase matching can only be achieved if a non negligible fractionγ_plof the gas is ionized and the plasma densityρ_pl=γ_plρ_atif sufficiently high. The fractionγ_plfor which phase matching is achieved is given by

γ_pl= 2ε₀m_eχ_atω²_l e²

≈0.04 forλ=800 nm and argon gas

If we include the Gouy phase for our experimental conditions the ionization radio re-duces toγ_pl ≈0.02 as the positive contribution of the Gouy phase adds to the positive contribution of the plasma refractive index.

11 19 27 35 43 51 59 Harmonic order

10⁻¹ 10⁰ 10¹

Coherencelength[m]

Figure 3.4:Simulated coherence length of a phase matched HHG source according to Eq. (3.3). Phase matching is achieved around the 27^thharmonic. The plasma dispersion, which is the only frequency dependent term, is sufficiently low so that for the full range the coherence length is larger than 0.1 m, i.e. phase matching is achieved.